Exercise 4.6.  Given a general complex polynomial equation P = 0, where polynomial P is equal to

and z and a₀...a_n are all complex expressions, show that (1) must be factorable into the following form:

Where b₁...b_n are also complex numbers.

     I guess one way to start is by showing that if b_x is a solution to (1), then any arbitrary (z-b_x) divides evenly into (1).  First substitute b_x into (1) and solve for a₀:

     Now divide P by z-b_x to arrive at the following:

     Substitute a₀ by the result in (3) to obtain:

     Which can be rearranged into:

     Or:

      Which shows that any z-b_x will divide evenly into all terms of the quotient in (6).  Now, using (3), 

      So z-b_x also divides into P.  Since this is true for any b_x, we can say that

      Q is the remainder which, when multiplied with the divisor D in (8), will provide a fully factored expression for P, i.e. QD = P.  To find Q, note that a sequence of products such as found in D can be expanded as follows:

      Where all A_x incorporate products and sums of the solutions b_x.  It's worth noting that all A_x as well as the final standalone constant (the product of all the b_x) correspond to the various A_x found in (1).  However (1) also has a factor a_n that is a coefficient for z_n.  a_n is therefore the missing factor Q needed to ensure that P perfectly factors Q. (DUBIOUS!)  Upon multiplying (9) by a_n, each coeffficient and the final standalone constant will equal each coefficient andd standalone constant in (1).  Therefore the final factorization for (1) is:

Shit's Messed Up

      Everything about this solution seemed to go smoothly until, unfortunately, the key moment when I had to figure out wtf happened to a_n.  I am not very happy that a_n did not arise naturally but instead had to be artificially multiplied in to (9) based on examination of (1): it seems too much like I am using my guilty knowledge of what the answer is supposed to be and then jerry-rigging the solution accordingly.  On the other hand, since z_n has a coefficient of 1 in (9), I suppose that you could always factor out the term a_n with the result that that coefficient becomes 1/a_n.  It still seems shitty and I'd be grateful if anyone knows how to go through these steps in a way that a_n pops up naturally as a coefficient for z_n.  Another potentially much more serious problem with the whole thing is that it doesn't use any of the properties of complex numbers.  Which is troubling.

     I apologize for the tininess of most of the inline math, it's ridiculously hard to read. Blogspot's image upload system is pretty crappy (as is their too-small text window, grah!).  But the only alternative I have is to basically buy  another box and start up my own, fully customizable blogging engine which would just take too much work.  Oh, if there was just a way to upload these things as inline-viewable PDFs!